System and method for an eddy-current field compensation in magnetic resonance imaging

ABSTRACT

A system and method for acquiring a calibrated eddy-current field model in magnetic resonance imaging (MM) are provided. The method may include one or more of the following operations. An eddy-current field model may be obtained. The eddy-current field model may be transformed by Laplace Transformation. Data of an eddy-current field may be obtained. The data of the eddy-current field may be processed. A calibrated eddy-current field model may be acquired. In addition, the calibrated eddy-current field model may be used to compensate an eddy-current field.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. application Ser. No.15/322,375 filed on Dec. 27, 2016, which is a U.S. national stage under35 U.S.C. § 371 of international Application No. PCT/CN2016/085132,filed on Jun. 7, 2016 and claims priority of Chinese Application No.201510527333.4 filed on Aug. 25, 2015, the entire contents of which areincorporated herein by reference.

TECHNICAL FIELD

The present disclosure generally relates to magnetic resonance imaging(MRI), and more particularly, to a system and method for an eddy-currentfield compensation in magnetic resonance imaging.

BACKGROUND

A magnetic resonance imaging (MRI) system may be used to diagnosemedical conditions by exploiting a powerful magnetic field and radiofrequency (RF) techniques. During the process of magnetic resonanceimaging (MRI), current in one or more gradient coils may change time,thus induces eddy-current around the conducting structures. Theeddy-current may cause a field varying spatially and/or temporally, andmay result in a distortion of the gradient magnetic field within theimaging region and/or degradation of the quality of the MRI image.

In order to compensate for the effects of the eddy-current, additionalcoils may be used as self-shielding coils for generating a countergradient magnetic field. The counter gradient magnetic field may reducethe interaction between coils and magnet to restrain eddy-current.However, residual eddy-current may remain.

SUMMARY

In a first aspect of the present disclosure, a magnetic resonanceimaging (MM) system is provided. In some embodiments, the MM system mayinclude an MRI scanner and a device. The device may include a processorand a computer-readable storage medium. The computer-readable storagemedium may store a computer program having instructions. Theinstructions, when executed by the processor, may cause the processor toperform operations. The processor may obtain an eddy-current fieldmodel. The eddy-current field model may be transformed by LaplaceTransformation. Data of an eddy-current field may be obtained. The dataof the eddy-current field may be processed based on the eddy-currentfield model to obtain a first parameter. A second parameter may beacquired by fitting the eddy-current field model based on the data andthe first parameter. A calibrated eddy-current field model may beacquired based on the second parameter.

In a second aspect of the present disclosure, a method is provided. Themethod may include one or more of the following operations. Aneddy-current field model may be obtained. The eddy-current field modelmay be transformed by Laplace Transformation. Data of a firsteddy-current field may be obtained. The data of the first eddy-currentfield based on the eddy-current field model may be processed to obtain afirst parameter. A second parameter may be acquired by fitting theeddy-current field model with the data and the first parameter. Acalibrated eddy-current field model based on the second parameter may beacquired.

In some embodiments, the eddy-current field may be compensated based onthe calibrated eddy-current field model.

In some embodiments, the data of the eddy-current field may be processedusing inverse Laplace transformation (ILT) to obtain a Laplacianspectrum. A peak of the Laplacian spectrum may be identified.Information of the peak may be acquired. The first parameter may bedetermined based on the information of the peak.

In some embodiments, the first parameter may be an initial parameterrelating to the eddy-current field model.

In some embodiments, the Laplacian spectrum may include a sparseLaplacian spectrum or a smooth Laplacian spectrum.

In some embodiments, the information of the peak may include at leastone type of position, boundary, or intensity.

In some embodiments, the peak may include a reliable spectrum peak.

In some embodiments, a first threshold may be obtained. The absolutevalue of intensity of the peak may be compared with the first threshold.The peak may be marked as a reliable spectrum peak if the absolute valueof the intensity of the peak is equal to or exceeds the first threshold.

In some embodiments, the first parameter may be determined based on acorresponding parameter of the reliable spectrum peak.

In some embodiments, a boundary of the reliable spectrum peak may bedetermined. A total intensity of the reliable spectrum peak within theboundary may be calculated. The first parameter based on the totalintensity of the reliable spectrum peak may be set.

In some embodiments, regularization may be applied to calculate the dataof the eddy-current field based on inverse Laplace transformation (ILT).

In some embodiments, a nonlinear least square fitting may be applied.

In some embodiments, the nonlinear least square fitting may include oneor more bilateral linear inequality constraints.

In some embodiments, the calibrated eddy-current field model may beacquired by performing a computer program stored in a non-transitorycomputer-readable storage medium.

In some embodiments, the computer program may have instructions thatwhen executed by the processor, the processor may perform operations toacquire the calibrated eddy-current model.

In some embodiments, the calibrated eddy-current field model may beacquired by a magnetic resonance imaging (MRI) system including an MRIscanner, a processor, and instructions.

In some embodiments, when the instructions are executed by theprocessor, the processor may perform operations to acquire thecalibrated eddy-current field model.

In some embodiments, at least one characteristic component of theeddy-current field may be identified based on the calibratededdy-current field model and further used to compensate the eddy-currentfield.

In some embodiments, a plurality of characteristic components of theeddy-current field may be identified based on the calibratededdy-current field model and further used to compensate the eddy-currentfield.

In some embodiments, a second eddy-current field including the pluralityof characteristic components may be obtained and compensated based onthe calibrated eddy-current field model.

Additional features will be set forth in part in the description whichfollows, and in part will become apparent to those skilled in the artupon examination of the following and the accompanying drawings or maybe learned by production or operation of the examples. The features ofthe present disclosure may be realized and attained by practice or useof various aspects of the methodologies, instrumentalities andcombinations set forth in the detailed examples discussed below.

BRIEF DESCRIPTION OF THE DRAWINGS

The present disclosure is further described in terms of exemplaryembodiments. These exemplary embodiments are described in detail withreference to the drawings. These embodiments are non-limiting exemplaryembodiments, in which like reference numerals represent similarstructures throughout the several views of the drawings, and wherein:

FIG. 1 illustrates a magnetic resonance imaging (MRI) system accordingto some embodiments of the present disclosure;

FIG. 2 is a block diagram illustrating the processing module accordingto some embodiments of the present disclosure;

FIG. 3 is a flowchart illustrating a process for eddy-current fieldcompensation according to some embodiments of the present disclosure;

FIG. 4 is a flowchart illustrating a process for eddy-current fieldcompensation according to some embodiments of the present disclosure;

FIG. 5 is a flowchart illustrating a process for acquiring aneddy-current field compensation model according to some embodiments ofthe present disclosure;

FIG. 6 is a flowchart illustrating a method of processing eddy-currentfield data according to some embodiments of the present disclosure;

FIG. 7 is a diagram illustrating Laplacian spectrums under differentintensities of punishment according to some embodiments of the presentdisclosure;

FIG. 8 is a flowchart illustrating a process for determining reliablespectrum peaks in a Laplacian spectrum according to some embodiments ofthe present disclosure;

FIG. 9 is a schematic diagram illustrating a Laplacian spectrumaccording to some embodiments of the present disclosure;

FIG. 10 is a flowchart illustrating a process for acquiring initialparameters according to some embodiments of the present disclosure;

FIG. 11 is a flowchart illustrating a process for determining theboundary of a reliable peak according to some embodiments of the presentdisclosure;

FIG. 12 is a diagram depicting a curve of raw eddy-current and a curveof fitting eddy-current according to some embodiments of the presentdisclosure; and

FIG. 13 is a diagram depicting curves that fit eddy-current of differentingredients according to some embodiments of the present disclosure.

DETAILED DESCRIPTION

In the following detailed description, numerous specific details are setforth by way of examples in order to provide a thorough understanding ofthe relevant disclosure. However, it should be apparent to those skilledin the art that the present disclosure may be practiced without suchdetails. In other instances, well known methods, procedures, systems,components, and/or circuitry have been described at a relativelyhigh-level, without detail, in order to avoid unnecessarily obscuringaspects of the present disclosure. Various modifications to thedisclosed embodiments will be readily apparent to those skilled in theart, and the general principles defined herein may be applied to otherembodiments and applications without departing from the spirit and scopeof the present disclosure. Thus, the present disclosure is not limitedto the embodiments shown, but to be accorded the widest scope consistentwith the claims.

It will be understood that the term “system,” “unit,” “module,” and/or“block” used herein are one method to distinguish different components,elements, parts, section or assembly of different level in ascendingorder. However, the terms may be displaced by other expression if theymay achieve the same purpose.

It will be understood that when a unit, module or block is referred toas being “on,” “connected to” or “coupled to” another unit, module, orblock, it may be directly on, connected or coupled to the other unit,module, or block, or intervening unit, module, or block may be present,unless the context clearly indicates otherwise. As used herein, the term“and/or” includes any and all combinations of one or more of theassociated listed items.

The terminology used herein is for the purposes of describing particularexamples and embodiments only, and is not intended to be limiting. Asused herein, the singular forms “a,” “an” and “the” may be intended toinclude the plural forms as well, unless the context clearly indicatesotherwise. It will be further understood that the terms “include,”and/or “comprising,” when used in this disclosure, specify the presenceof integers, devices, behaviors, stated features, steps, elements,operations, and/or components, but do not exclude the presence oraddition of one or more other integers, devices, behaviors, features,steps, elements, operations, components, and/or groups thereof.

FIG. 1 illustrates a magnetic resonance imaging system according to someembodiments of the present disclosure. As illustrated, an MRI system 100may include an MRI scanner 110, a processing module 120, and a displaymodule 130. The MRI system's image include a main magnet 111, aprocessor and a computer-readable storage medium 121 are presented. TheMRI scanner 110 may include a magnet module 112 and a radio frequency(RF) module 113. The magnet module 112 may include a main magnet filedgenerator and/or a plurality of gradient coils. The main magnet fieldgenerator may create a static magnetic field BO during an MRI process.The main magnetic field generator may be of various types including, forexample, a permanent magnet, a superconducting electromagnet, aresistive electromagnet, etc. The gradient coils may include X-gradientcoils, Y-gradient coils, and Z-gradient coils. The gradient coils maygenerate magnetic field gradients to the main magnetic field in the X,Y, and/or Z directions to encode the spatial information of the subjectlocated in the MRI scanner 110. In some embodiments, the X-gradient mayprovide the X-position information, which may be known as frequencyencoding. In some embodiments, the Y-gradient may provide the Y-positoninformation, which may be known as phase encoding. The RF module 113 mayinclude RF transmitting coils and/or receiving coils. These RF coils maytransmit RF signals to or receive RF signals from a subject of interest.In some embodiments, the function, size, type, geometry, position,amount, and/or magnitude of the magnet unit 112 and/or of the RF unit113 may be determined or changed according to one or more specificconditions. For example, according to the difference in function andsize, the RF coils may be classified as volume coils and local coils. Insome embodiments, the volume coils may include birdcage coils,transverse electromagnetic coils, surface coils, saddle coils, etc. Insome embodiments, the local coils may include birdcage coils, solenoidcoils, saddle coils, flexible coils, etc.

The processing module 120 may process different kinds of instructionsreceived from different modules. For further understanding the presentdisclosure, several examples are given below, but the examples do notlimit the scope of the present disclosure. For example, in someembodiments, the processing module 120 may process data of the gradientmagnetic field received from the magnet module 112 and calculated one ormore parameters based on these datum and fitting the eddy-current fieldmodel with the parameters. The display unit 130 may receive input and/ordisplay output information. The input and/or output information mayinclude programs, software, algorithms, data, text, number, images,voice, or the like, or any combination thereof. For example, a user oran operator may input some initial parameters or conditions to initiatea scan. As another example, some information may be imported fromexternal resource, such as a floppy disk, a hard disk, a wirelessterminal, or the like, or any combination thereof.

It should be noted that the above description of the MRI system 100 ismerely provided for the purposes of illustration, and not intended tolimit the scope of the present disclosure. For persons having ordinaryskills in the art, multiple variations and modifications may be madeunder the teachings of the present disclosure. For example, the assemblyand/or function of the MRI system 100 may be varied or changed accordingto specific implementation scenarios. Merely by way of example, someother components may be added into the MRI system 100, such as a patientpositioning unit, a gradient amplifier unit, and other devices or units.Note that the Mill system may be a traditional or a single-modalitymedical system, or a multi-modality system including, e.g., a positronemission tomography-magnetic resonance imaging (PET-MRI) system, acomputed tomography-magnetic resonance imaging (CT-MRI) system, a remotemedical MRI system, and others, etc. However, those variations andmodifications do not depart from the scope of the present disclosure.

FIG. 2 is a block diagram illustrating the processing module 120according to some embodiments of the present disclosure. The processingmodule 120 as illustrated in FIG. 1 may process information before,during, or after an imaging procedure. Note that the construction of theprocessing module 120 may have some other variations, and that FIG. 2 isprovided for illustration purposes. The processing module 120 mayinclude a CPU. The CPU may be a central processing unit (CPU), anapplication-specific integrated circuit (ASIC), an application-specificinstruction-set processor (ASIP), a graphics processing unit (GPU), aphysics processing unit (PPU), a microcontroller unit, a digital signalprocessor (DSP), a field programmable gate array (FPGA), an ARM, or thelike, or any combination thereof. As shown in FIG.2, the processingmodule 120 may include a collection unit 201, a modeling unit 202, acomputation unit 203 and a compensation unit 204.

The collection unit 201 may obtain different kinds of information. Theinformation may include information about the Mill scanner 110, themagnet module 112, the RF module 113, or the like, or any combinationthereof. In some embodiments, the information may be a subject position(e.g., within an Mill system), the main and/or gradient magnetic fieldintensity, the data of the eddy-current field (or referred to as theeddy-current field data), the radio frequency phase and/or amplitude,and so on. The information may include information from a user and/orother external resource. Exemplary information from a user may includethe parameters regarding the parameters of the main and/or gradientmagnetic field, a subject of interest (e.g., the type of tissue to beimaged, etc.), slice thickness, an imaging type, a spin echo type (e.g.,spin echo, fast spin echo (FSE), fast recovery FSE, single shot FSE,gradient recalled echo, fast imaging with stead-state procession, and soon), a flip angle value, acquisition time (TA), echo time (TE),repetition time (TR), echo train length (ETL), the number of phases, thenumber of excitations (NEX), inversion time, bandwidth (e.g., RFreceiver bandwidth, RF transmitter bandwidth, etc.), or the like, or anycombination thereof. In some embodiments, the information may be aneddy-current field model, one or more parameters of the eddy-currentfield model including, e.g., one or more parameters for a model for afirst-stage eddy-current field compensation based on prior experience,etc.

The modeling unit 202 may obtain a model of the eddy-current field. Insome embodiments, the modeling unit 202 may obtain the eddy-currentfield model from a user and/or other external resource. Exemplary modelsmay be a Multiple-index model, etc. In some embodiments, the modelingunit 202 may perform Laplace transformation on the eddy-current fieldmodel. In some embodiments, the modeling unit 202 may obtain thecalibrated eddy-current field model based on the parameters from thecomputation unit 203. In some embodiments, the modeling unit 202 mayobtain a calibrated eddy-current field model based on optimizedparameters from the computation unit 203.

The computation unit 203 may calculate different kinds of information.In some embodiments, the computation unit 203 may calculate the data ofthe eddy-current field to obtain the parameters of an eddy-current fieldmodel. In some embodiments, the computation unit 203 may further performcurve-fitting based on the eddy-current field model and the data of theeddy-current field to obtain the calibrated eddy-current field model. Insome embodiments, the computation unit 203 may transform the data of theeddy-current field using inverse Laplace transformation (ILT) to obtaina Laplacian spectrum. In some embodiments, the computation unit 203 mayidentify a peak of the Laplacian spectrum. In some embodiments, thecomputation unit 203 may compare the absolute value of intensity of thepeak with a first threshold. In some embodiments, the computation unit203 may identify a reliable spectrum peak, if the absolute value ofintensity of which is not less than the first threshold. In someembodiments, the computation unit 203 may determine a boundary of thereliable spectrum peak. In some embodiments, the computation unit 203may calculate a total intensity of the reliable spectrum peak within theboundary. In some embodiments, the computation unit 203 may applyregularization to calculate the data of the eddy-current field based oninverse Laplace transformation (ILT). In some embodiments, thecomputation unit 203 may apply a nonlinear least square fitting. In someembodiments, the computation unit 203 may apply bilateral linearinequality constraints.

The compensation unit 204 may compensate an eddy-current field. In someembodiments, the compensation unit 204 may compensate the eddy-currentbased on an empirical value from a user and/or another externalresource. In some embodiments, the compensation unit 204 may compensatethe eddy-current field based on the calibrated eddy-current field model.In some embodiments, the compensation unit 204 may compensate theeddy-current field based on the gradient magnetic field.

It should be noted that the above description of the MRI system ismerely provided for the purposes of illustration, and not intended tolimit the scope of the present disclosure. For persons having ordinaryskills in the art, multiple variations and modifications may be madeunder the teaching of the present invention. For example, the assemblyand/or function of processing module may be varied or changed. However,those variations and modifications do not depart from the scope of thepresent disclosure.

FIG. 3 is a flowchart illustrating a process for eddy-current fieldcompensation according to some embodiments of the present disclosure.Eddy-current field data may be acquired in step 301. In someembodiments, the eddy-current field data may be acquired by taking ameasurement when the MRI scanner operates. In different operationalsituations, different gradient magnetic fields may be desired, anddifferent eddy-current fields may be generated according to theoperation parameters. Different eddy-current field data may becategorized into several groups corresponding to their operationalparameters. Exemplar operational parameters may include currentintensity, duration time, or the like, or a combination thereof.Additionally or alternatively, eddy-current field data may be differentunder the same operational parameters due to differences in, forexample, material(s) of one or more magnetic conductive components, theconfiguration of gradient coils, or the like, or a combination thereof.For example, in the magnet module 112, the length of a magneticconductive component may be longer than another in a different Millscanner. As another example, the gaps of the magnetic conductivecomponents in an Mill scanner may be different from those in anotherMill scanner. In some embodiments, the eddy-current field data may alsobe set by default. For example, in a simulation experiment, theeddy-current field data may be set by a user according to priorexperience.

In step 302, a mathematical model may be selected. The number of theselected mathematical models may be one, or two, or any positiveinteger. The mathematical models may be stored in a mathematical modeldatabase. In some embodiments, the mathematical model database may bestored on a local device, for example, in the processing module 120 or astorage medium of the magnetic resonance imaging system. In someembodiments, the mathematical model database may be stored in a server(e.g., a cloud server) (not shown in the drawings).

A mathematical model may include or be based on an exponential function,a multi-exponential function, an attenuation model, or the like, or acombination thereof. For different MRI scanners, a same mathematicalmodel or different mathematical models may be selected. Different Millscanners may include MRI scanners of different designs, by differentmanufacturers, etc. Merely by way of example, two MRI scanners of a samedesign by a same manufacture, when operating according to sameoperational parameters may generate different eddy-current fields; asame mathematical model or different mathematical models may beselected. As another example, a same mathematical model may be used inconnection with multiple MRI scanners of a same design or differentdesigns, by a same manufacturer or different manufacturers, etc. Theselected mathematical model may contain one or more unknown parameters,which may be determined in subsequent steps. Otherwise, if the selectedmathematical model does not contain unknown parameters, it may be usedin the compensation for eddy-current field.

In step 303, unknown parameters of the selected mathematical model maybe determined according to the eddy-current field data of an MRIacquired previously. The calculation may be performed in the computationunit 203. In some embodiments of the present disclosure, thedetermination may be based on all or part of the eddy-current fielddata. In some embodiments, the determination may be optimized based onone or more characteristics of the eddy-current field data. Theoptimization may include improvements in terms of the time that it maytake to determine one or more unknown parameters of the selectedmathematical model, the result of the compensation, or the like, or acombination thereof. In some embodiments, the unknown parameter(s) ofthe selected mathematical model may be determined based on theeddy-current field data and some other parameters of the MRI scanner.For example, the RF coil(s) or some other components in the Mill scannermay affect the gradient magnetic field. The parameters of an RF coil orsome other components may be taken into consideration duringcalculation. As used herein, a mathematical model in combination withdetermined parameter(s) may be referred to as a calibrated mathematicalmodel. A calibrated mathematical model may provide guidance foreddy-current field compensation. As used herein, a mathematical modelfor eddy-current field compensation may be referred to as a compensationmodel. As used herein, a calibrated mathematical model for eddy-currentfield compensation may be referred to as a calibrated compensationmodel. In some embodiments, because the unknown parameter(s) of themathematic model are determined using data relating to a specific MRIscanner, the corresponding calibrated mathematical model may be used foreddy-current field compensation for the specific MRI scanner. Differenteddy-current fields may be generated in a specific Mill scanner underdifferent operation conditions. For instance, the ratio of thezero-order and/or the first-order eddy-currents to higher ordereddy-currents may be changed with the changing of operation conditions.A calibrated mathematical model may be determined according to the kindof eddy-current field generated in the gradient magnetic field.Different calibrated mathematical models may be determined according todifferent eddy-current fields generated in the gradient magnetic field.One or more calibrated mathematical models may be stored locally andretrieved when needed.

In step 304, the eddy-current field in the gradient magnetic field maybe compensated. The calibrated mathematical model may be retrieved for aspecific Mill scan with a set of operational parameters using an MRIscanner. The computation unit 203 may calculate a simulated eddy-currentfield based on the calibrated mathematical model. The calculation of thesimulated eddy-current field may also be based on at least some of theoperational parameters. The simulated eddy-current field may be anapproximation of an eddy-current generated in an MRI scan. The simulatededdy-current field may be then transformed into a control signal and/orfurther transferred into compensation hardware. Exemplary compensationhardware may include a digital filter, etc. The compensation hardwaremay generate a compensation eddy-current field that is opposite to thesimulated one. The actual eddy-current field generated by the gradientcoils of the MRI scanner may be compensated or cancelled out by thecompensation eddy-current field.

It should be noted that FIG. 3 is for illustration purposes, notintended to limit the scope of the present disclosure. For example, thestep 302 may be performed before step 301. In some embodiments, amathematical model may be selected such that data related to aneddy-current field may be acquired according to the selectedmathematical model for further processing.

FIG. 4 is a flowchart illustrating a process for eddy-current fieldcompensation according to some embodiments of the present disclosure.The compensation may be separated into two stages. The first stage maybe a coarse compensation; the second stage may be an accuratecompensation. A coarse compensation of the eddy-current field in agradient magnetic field may be achieved using a self-shielding coilsurrounding a gradient coil. The current direction in the self-shieldingcoil may be opposite to the gradient coil. Parameters of theself-shielding coil may be adjusted to improve the eddy-current fieldcompensation. Exemplary adjustable parameters of the self-shielding coilmay include the length of the self-shielding coil, the perimeter of theself-shielding coil, the current intensity applied through theself-shielding coil, or the like, or a combination thereof. As anotherexample, coarse compensation may be achieved using aneddy-current-resisting board made of a high electrical resistivitymaterial. Parameters of the eddy-current-resisting board may be adjustedto improve the eddy-current field compensation. Exemplary adjustableparameters of the eddy-current-resisting board may include a geometrysize of the eddy-current-resisting board, the position of theeddy-current resisting board relative to the gradient coil, the patternsof groove on the surface of the eddy-current resisting board, or thelike, or a combination thereof. In step 401, a coarse compensation maybe executed. At least a part of the eddy-current field may becompensated or cancelled out. However, residual eddy-current may stillexist.

In step 402, an assessment may be performed to determine whether thecoarse compensation satisfies a criterion. After the coarse compensationin step 401, the residual eddy-current after the first stageeddy-current field compensation (or referred to as “first residualeddy-current” as illustrated in FIG. 4) may be measured. If the firstresidual eddy-current exceeds a threshold (“first threshold” asillustrated in FIG. 4), the coarse compensation may be consideredinsufficient. Insufficient coarse compensation may cause morecomputational costs in a second stage compensation. The first thresholdmay be determined based on, for example, prior experience, the capacityand/or costs (e.g., computational costs, etc.) of the second stagecompensation, or the like, or a combination thereof.

For example, according to prior experience, a certain amount of residualeddy-current may be acceptable, and a resultant MRI image may besufficient for diagnosis purposes. As another example, a certain amountof residual eddy-current is within the compensation capacity of thesecond stage compensation so that the residual eddy-current may becompensated in the second stage compensation and a resultant image maybe acceptable or sufficient for diagnosis purposes. As a furtherexample, the cost (e.g., computational cost, etc.) of compensating acertain amount of residual eddy-current may be acceptable. The thresholdmay be set based on such an amount of residual eddy-current describedabove.

Based on the assessment, if the first residual eddy-current is beyondthe threshold (e.g., “first threshold” as illustrated in FIG. 4), theprocess may proceed to step 401 to adjust the coarse compensation. Ifthe first residual eddy-current is below the first threshold, a secondstage compensation may be performed. The second stage compensation maybe an accurate compensation.

In step 403, a second stage eddy-current field compensation may beperformed. A mathematical model may be determined in this step andfurther used to compensate the eddy-current field. Exemplary methods ofdetermining a mathematical model and/or compensating the eddy-currentfield according to the mathematical model may be disclosed elsewhere inthe present disclosure (e.g., that illustrated in FIG. 5). At the secondstage compensation, the eddy-current field may be further compensated.In step 404, the performance of the second stage compensation may beassessed. A measurement may be performed to collect the informationabout the residual eddy-current that remains after the second stagecompensation. In some embodiments of the present disclosure, when aresidual eddy-current after the second stage eddy-current fieldcompensation (or referred to as “second residual eddy-current” asillustrated in FIG. 4) exceeds a threshold (e.g., “second threshold” asillustrated in FIG. 4), the process may proceed to step 403 to performfurther second stage eddy-current field compensation in a different way.For example, a new mathematical model or a previously determinedmathematical model may be used. In some embodiments, the parameters ofthe previously determined mathematical model may be adjusted or revised.If the second residual eddy-current is below the second threshold, theprocess may proceed to step 405 indicating that the eddy currentcompensation may be considered sufficient. The first threshold may bedifferent from the second threshold.

It should be noted that the process illustrated in FIG. 4 is forillustration purposes, and not intended to limit the scope of thepresent disclosure. Merely by way of example, the coarse compensationstep 401 and/or the corresponding assessment step 402 may be omitted. Asanother example, when the first stage compensation is performed, theassessment step 402 may be omitted. For example, a self-shielding coiland its parameters may be determined based on prior experience. Theassessment step may be executed for several rounds of experiments and astandard parameters of a self-shielding coil may be determined. In thefirst stage compensation, the performance of compensation result may beassumed to be acceptable and need no test. Similar to step 402, theassessment of step 404 may be omitted. For example, if the compensationmodel determined in step 403 is sufficient based on numerous times ofexperiments, or the deviation of an MRI image caused by residualeddy-current is not obvious when applying said compensation model, theperformance of compensation result may be assume to be acceptable, andthe assessment in step 404 may be omitted.

FIG. 5 is a flowchart illustrating a process for acquiring aneddy-current field compensate model according to some embodiments of thepresent disclosure. In step 501, an eddy-current field model may beobtained. Merely by way of example, according to estimation or one ormore prior eddy-current measurements, an eddy-current field may beapproximated as a superposition of several exponential attenuationcurves. Accordingly, an exemplary mathematical model may be amulti-exponential attenuation model. The eddy-current field model mayinclude a mathematical model of this or other forms. The eddy-currentfield may be described using an eddy-current field model according tothe mathematical model and information regarding the eddy-current field.The eddy-current field model may be transformed into anotherexpressions. Exemplary transformation may include Laplacetransformation, Z-transformation, Fourier transformation, or the like,or a combination thereof. In some embodiments of the present disclosure,the multi-exponential attenuation model may be transformed using Laplacetransformation as:

d _(m)=∫₀ ^(∞) x(k)e ^(−T) ^(m) ^(·k) dk   (1)

where

${x = {\gamma \frac{G_{x}}{T_{A}}{{\alpha\tau}( {1 - e^{{- T_{A}}/\tau}} )}}},$

k=1/τ, d_(m) may represent eddy-current data of m, γ may represent agyromagnetic ratio of a proton, G_(x) may be a testing gradient, T_(A)may represent a duration time of a falling edge of a testing gradientpulse, T_(m) may represent a delay of the eddy-current data. As usedherein, the delay may refer to a time interval between the end time of atesting gradient and the time that a corresponding eddy-current ismeasured or detected. The corresponding eddy-current may be generated bythe testing gradient. m(m=1,2, . . . , M) represents the sequence numberof sampling timing, a may represent the amplitude of an eddy-current,and τ may represent a characteristic time of an eddy-current component(or referred to as a characteristic component described below). Theeddy-current filed may include one or more eddy-current components. Eacheddy-current component may correspond to a characteristic time τ. Thelinear part x(k) may be expressed as:

${x = {\gamma \frac{G_{x}}{T_{A}}{{\alpha\tau}( {1 - e^{{- T_{A}}/\tau}} )}}},$

and the nonlinear part k may be expressed as: k=1/τ.

Equation (1) may indicate that d_(m) is the Laplace transformation ofx(k). In some embodiments, x(k) may be obtained by inverse Laplacetransformation of d_(m). It should be noted that, the examples shownabove are merely for illustration purposes, and not intended to limitthe scope of the present disclosure. In some embodiments, when thelinear part and the nonlinear part of a mathematical model areindependent and the expression of the mathematical model conforms toLaplace transformation, the mathematical model may also be applicable orsuitable.

In step 502, data of an eddy-current field may be acquired. The measureddata may be further used to calculate the calibrated eddy-current fieldmodel. In some embodiments of the present disclosure, the mathematicalmodel determined before may contain some undetermined parameters, whichneed to be calculated using the measured data. Therefore, after step502, the process may proceed to 504, when the measured data do not needfurther process. For example, the initial parameters of the obtainededdy-current field model may be measured directly by taking someexperiments. In some embodiments, the measured data may be processed instep 503. The processing in step 503 may be performed manually by, forexample, a user, or by a computation unit or a hardware. Exemplaryprocessing applicable in step 503 may include inverse Laplacetransformation, z-transformation, Fourier transformation, or the like,or a combination thereof. In step 504, the undetermined parameters ofthe eddy-current field model may be calculated according to the measureddata of eddy-current and/or the processed measured data of eddy-currentfield. One or more initial parameters of the eddy-current field modelmay be obtained.

In step 505, a calibrated eddy-current field model may be determinedaccording to the calculation of the undetermined parameters thereof.Details to calibrate the eddy-current field model may be disclosedelsewhere in the present disclosure.

FIG. 6 is a flowchart illustrating an exemplary method for processingeddy-current field data to obtain one or more initial parameters of aneddy-current field model according to some embodiments of the presentdisclosure. In step 601, a set of eddy-current field data may beacquired as described elsewhere in the present disclosure. See, forexample, the description regarding step 301 in FIG. 3.

The acquired eddy-current field data may be processed in step 602. Themethod to process may include inverse Laplace transformation. In someembodiments, the inverse Laplace transformation may be similar todiscrete Fourier transformation and realized by calculating a linearinversion. In some embodiments, the inverse radical may be discretized.In some embodiments, the discrete values of k may cover the entire orpart of the range of values. The entire range may be the range of widthof the corresponding Laplacian spectrum. In some embodiments, the valuesof k may cover one or several orders of magnitude. The values of k maybe set at an equal logarithm interval. The application of the inverseLaplace transformation may acquire corresponding Laplacian spectrum. Thesetting of the k values may affect the acquired Laplacian spectrum. Forexample, if the number of the k values is too few, the number of thepeaks of the Laplacian spectrum may be too few and not acceptable forfurther processing. If the number of the k values is too many, thesubsequent processing may consume a large amount of computationresource. In some embodiments, a several dozen to several hundred ofdiscrete k values may be acquired. Assuming that n(n=1,2, . . . , N)represents the sequence of k, Equation (1) may be rewrote as:

d _(m)=Σ_(n=1) ^(N) x _(n) e ^(−T) ^(m) ^(·k) ^(n) .   (2)

Equation (2) may be expressed in the following form:

d=Cx,   (3)

where both d=[d₁, d₂, . . . , d_(m)]^(T) and x=[x₁, x₂, . . . ,x_(n)]^(T) may represent column vectors. Superscript T represents atransposition. C is a coefficient matrix of m rows*n columns. In someembodiments, C may be a Laplacian operator that may be defined as:

C _(m,n) =e ^(−T) ^(m) ^(·k) ^(n) .   (4)

In some embodiments, Equation (3) may be underdetermined (m<n), so thatthe result of Equation (2) may be ill conditioned while x=C/d and/orx=C⁺d. In some embodiments, superscript+may represent a Moore-Penrosegeneralized inverse. A regularization method may be used to solve theill conditioned result. In some embodiments, the discrete values of kmay be substituted into Equation (5) shown below to calculate x, and thedistribution function of x relating to k may be as follows:

$\begin{matrix}{{\min\limits_{x}\{ {{{{Cx} - d}}_{2}^{2} + {R(x)}} \}},} & (5)\end{matrix}$

where R (x) may be a penalty term for supplementing a specificconstraint condition. Exemplary constraint conditions may includesmoothness, sparsity, or the like, or a combination thereof. Fordifferent kinds of constrain conditions, the number of eddy-currentcomponents shown in the Laplacian spectrum may be different. Merely byway of example, the number of eddy-current components is limited; asparse constraint condition may be used. An l₁-norm punishment asfollows is an exemplary sparse constraint condition:

R(x)=λ∥x∥ ₁,   (6)

where λ may represent a Lagrange multiplier, and/or a regularizationcoefficient. In some embodiments, the regularization coefficient may beused to adjust the punishment intensity. In some embodiments, theregularization coefficient may keep a balance between the regressioneffect and sparseness. An appropriate value of λ may be acquired with aposterior examination strategy. Exemplary posterior examinationstrategies may include an L-curve method, limiting the sparseness ofspectrum peaks, or the like, or a combination thereof. An l₁-normpunishment may provide a mandatory sparseness. In some embodiments, anl₁-norm punishment may be used in the construction of a sparse Laplacianspectrum of x relating to k. For example, the sparse Laplacian spectrummay be acquired using an l₁-norm punishment as disclosed in “Sparse MRI:The Application of Compressed sensing for Rapid MR Imaging, M. Lustig,D. Donoho, and J. M. Pauly, Magn. Reson. Med., 2007, 58, 1182,” thecontents of which are hereby incorporated by reference.

In step 603, a Laplacian spectrum may be acquired according to theprocess in step 602. FIG. 7 is a diagram illustrating various Laplacianspectrums acquired using different punishment intensities according tosome embodiments of the present disclosure. For different λ, the numbersof peaks in the Laplacian spectrums may be different. For instance, whenλ=10,000, the acquired Laplacian spectrum may have one peak. As thevalue of λ decreases, the number of peaks of a Laplacian spectrum mayincrease, and the intensity of a peak of the Laplacian spectrum mayincrease. The acquired Laplacian spectrum may be further processed toextract relevant information. An appropriate value of λ and an image ofa corresponding Laplacian spectrum may be selected. In some embodiments,the selection may be based on knowledge from, for example, priorexperience. In some embodiments, the selection may be based on thecomplexity of subsequent processing steps. For example, thecorresponding Laplacian spectrum may be determined based on the value ofpeak intensity. In some embodiments, information of the peaks of aLaplacian spectrum may be extracted. Exemplary peak information mayinclude location, intensity, or the like, or a combination thereof. Theextracted information may be used to further calculate the unknownparameters of the obtained eddy-current field model. Parameters a and iof the eddy-current field model may be acquired based on x and k. Insome embodiments, at least some of the values of x may be non-zero. Insome embodiments, the values of k may be related to the values of x. Theeddy-current field model with the parameters so determined may be usedin compensation for the eddy-current field. The raw data of an actualeddy-current field may include noise, and the acquired sparse Laplacianspectrum may contain pseudo-components in a small or large amount,thereby affecting the quality of an image based on the raw data. Thevalue of a discrete inverse radical relating to a characteristiccomponent may be within a range of the characteristic component of theeddy-current field. As used herein, the characteristic component of aneddy-current field may be at least one of the eddy-current components ofthe eddy-current field that may describe or characterize theeddy-current field. The acquired Laplacian spectrum may be furtherprocessed.

Referring back to FIG. 6, in step 604, one or more reliable spectrumpeaks may be acquired by processing the Laplacian spectrum according tosome embodiments of the present disclosure. The reliable spectrum peakmay be acquired based on the intensity of a peak, or the location of apeak, or the like, of a combination thereof. In some embodiments, thereliable spectrum peak may be determined based on non-overlapping peaks.In some embodiments, the peaks overlapping may be determined not to bereliable spectrum peaks. In some embodiments, two peaks may bedetermined as one reliable peak. Details to acquire the reliablespectrum peaks according to some other embodiments may be foundelsewhere in the present disclosure.

In step 605, initial parameters of the obtained eddy-current field modelmay be calculated based on the reliable spectrum peaks. The initialparameters may be one or more sets of data. In some embodiments, thedata may be initial estimation data used in nonlinear least squarefitting of Equation (1). In some embodiments, the initial parameters maybe set as the values of x corresponding to one or more reliable peaks(or referred to as reliable spectrum peaks), and the correspondingvalues of k of the values of x. Details regarding the determination ofthe initial parameters may be found elsewhere in the present disclosure.

FIG. 8 is a flowchart illustrating a process for determining reliablespectrum peaks in a Laplacian spectrum according to some embodiments ofthe present disclosure. In step 801, a Laplacian spectrum may beacquired. Information of the peaks (or referred to as peak information)may be extracted from the spectrum. Exemplary peak information mayinclude location, intensity, or the like, or a combination thereof.Other characteristics of the peaks may be extracted and/or used insubsequent processing. In step 802, a spectrum peak may be retrievedaccording to at least some of the peak information. In some embodiments,the retrieval may be based on the location of each peak. For example,the peaks may be sorted and numbered by their locations. In someembodiments, the retrieval may start from the first one. In someembodiments, the retrieval may start from the last one. In someembodiments, all or some the peaks in a Laplacian spectrum may beretrieved. For example, every other peaks of a Laplacian spectrum may beretrieved.

The retrieved peaks may be analyzed in step 803. For instance, thecharacteristic feature of the peak may be calculated in step 803 asillustrated in FIG. 8. The characteristic feature may be the maximumintensity of a peak (also referred to as characteristic intensity), orthe integral of the corresponding area underneath the intensitydistribution of the peak. In step 804, an assessment may be performed todetermine whether a peak is a reliable peak. In some embodiments, if theabsolute value of the characteristic intensity of a peak is beyond apredetermined threshold, the peak may be considered as a reliable peak.

FIG. 9 is a schematic diagram illustrating a Laplacian spectrumaccording to some embodiments of the present disclosure. The peak A andpeak B may be representative peaks in a Laplacian spectrum. The twohorizontal dash lines may represent the predetermined threshold. Forexample, the absolute value of characteristic intensity of the peak A isbelow the dash line and that of the peak B is above the dash line.

Based on the assessment of step 804, if the absolute value of thecharacteristic intensity beyond a threshold, the process may proceed tostep 805. In step 805, the peak is satisfied the step 804 may be markedas a reliable peak. In connection with the exemplary peaks illustratedin FIG. 9, the peak B may be marked as a reliable peak, while peak A isnot marked as a reliable peak in step 805. In some embodiments, if thepeak is marked as a reliable peak, the information thereof may be storedin an area other different from where the information of an unreliablepeak is stored. In some embodiments, the reliable peak may be assigned alabel and may be used in further processing.

After the reliable peak is marked in step 805, the process may furtherproceed to step 806. If the peak does not satisfied the criterion ofstep 804, the process may proceed to step 806 without marking the peak.In some embodiments, an unreliable peak may be ignored in furtherprocessing. In step 806, another assessment may be performed todetermine whether the retrieval of peaks of a spectrum for assessment instep 804 is complete. In some embodiments, the retrieval may be stoppedwhen all the peaks in a spectrum image are retrieved for analysis. Insome embodiments, only some peaks of a spectrum need to be retrieved.Merely by way of example, every other peaks of a spectrum need to beretrieved, or one every three or four peaks of a spectrum need to beretrieved. For example, after a certain percentage of peaks areretrieved, the retrieval may terminate. As another example, when acertain number of reliable peaks are marked, the retrieval mayterminate. The certain number of reliable peaks may be enough forfurther processing. In step 807, the reliable peaks may be acquired. Bythis procedure as illustrated in FIG. 8, at least some of thepseudo-components caused by noise of the original data may be removed.

FIG. 10 is a flowchart illustrating a process for acquiring initialparameters according to some embodiments of the present disclosure. Instep 1001, a set of reliable spectrum peaks may be acquired. In someembodiments, the set of reliable spectrum peaks may be identified asdescribed elsewhere in the disclosure. See, for example, FIG. 8 and thedescription thereof. In step 1002, the reliable peaks may be retrieved.In some embodiments, the retrieval may start from the first reliablepeak to the last one based on the marked or numbered reliable peaks. Insome embodiments, a different order of the retrieval may be employed.

In step 1003, the boundary of the retrieved reliable peak in step 1002may be determined. The boundary of a reliable peak may be used tocalculate the total intensity of the spectrum peak. Referring to FIG. 9,the vertical dash lines a and b may be the boundaries of the peak D.

Based on the boundaries of a reliable peak, a specific area may bedetermined in step 1004. The specific area may represent the totalintensity of the reliable peak. For example, the total intensity may beset as the integral value of the specific area. In some embodiments, thetotal intensity of a peak may be set as the integral value of thespecific area defined by the intensity distribution of a reliable peak,the boundaries of the reliable peak, and the threshold intensity (e.g.,indicated by the dashed lines in FIG. 9). In step 1005, informationincluding the total intensity of the reliable peak may be acquired. Instep 1006, an assessment may be performed for determining whether theretrieval of reliable peaks are completed. In some embodiments, theretrieval may terminate when all the reliable peaks are retrieved. Basedon the assessment, if the retrieval is complete, the process may proceedto step 1007. Otherwise, the process may proceed to step 1002 toretrieve more reliable peaks.

In step 1007, the initial parameters may be determined according to atleast some of the peak information of the reliable peaks. In someembodiments, the values of k relating to the maximum values of x may beacquired. For example, for a reliable peak l, the value of k_(l)corresponding to the maximum value of x_(l) may be set as k0 _(l), andthe total intensity as x0 _(l). The peak information including thevalues of x0 _(l) and k0 _(l) corresponding to the reliable peak l or aplurality of the reliable peaks may be acquired to provide the sets of[x0 _(l)] and [k0 _(l)]. The initial parameters may be set as [x0 _(l)]and [k0 _(l)].

FIG. 11 is a flowchart illustrating a process for determining theboundary of a reliable peak according to some embodiments of the presentdisclosure. In step 1101, a reliable peak may be acquired. In someembodiments, the reliable peak may be numbered with a designation l. Instep 1102, the value of k_(l) corresponding to the characteristicintensity may be determined. As another example shown in FIG. 9, theposition of k_(l) may be determined based on the vertical dash line e inthe middle of the peak D. After the position of k_(l) is determined, theprocess may proceed to step 1103 and/or 1107.

In step 1103, the retrieval may start form k_(l) towards the directionof zero. For a given k, the corresponding absolute value of intensitymay be determined. In step 1104, an assessment may be performed todetermine whether the absolute value of the intensity is equal to orexceeds a threshold. If the absolute value of the intensity falls belowthe threshold, the process may proceed to step 1106, and the firstboundary may be determined. For example, referring to FIG. 9, the peak Dmay be a reliable peak according to some embodiments of the presentdisclosure. In FIG. 9, the retrieval may start from k_(l). k_(l) may bedenoted by the dash line in the middle of the peak D. As the retrievalstart from k_(l) towards the direction of zero, in step 1104, theretrieval may terminate at the point of α (set as left boundaryklb_(l)). In another word, the dash line at the point α is the firstboundary of the peak D. It should be noted that the threshold used todetermine the boundary may be the same as that used in the determinationof reliable peak. In some embodiments, the threshold used to determinethe boundary may be different from which used in the determination ofreliable peak. For example, the threshold may be a value relating to thearea located within the boundary beyond a certain percent of the area ofthe peak.

In step 1104, if the absolute value of the intensity exceeds thethreshold, a further assessment may be performed to determine if thespecific reliable peak overlaps with another reliable peak. In step1105, the absolute value of the intensity at a k value may be comparedwith the absolute value of the intensity at a preceding k value in thepreceding retrieval. In some embodiments, the retrieval may start fromthe maximum intensity of a reliable peak (e.g., the characteristicintensity of the peak), and the absolute value of the intensitycorresponding to the k values may decrease for the intensities retrievedsubsequently. If the absolute value of the intensity increase, there maybe two or more overlapping reliable peaks. If the corresponding absolutevalue of intensity exceeds the threshold and larger than the previousone in the last retrieval, the retrieval may terminate and the firstboundary may be determined. Otherwise, the assessment may proceed tostep 1103 for next retrieval. For example, as is shown in FIG. 9, thepeak C may include two overlapped peaks. The dash line at the point emay be the first boundary of the peak on the right side of the peak C,and the dash line at the point f may be the first boundary of the peakon the left side of the peak C according to the assessment of step 1104and step 1105.

In some embodiments, the retrieval may start form k_(l) towards thedirection of the positive infinity. For example, according to the rightbranch of the process in FIG. 11, the second boundary of a reliable peakmay be determined following the same steps as in the determination ofthe first boundary. As another example, in FIG. 9, the dash line atpoint b (set as right boundary kub_(l)) may be the second boundary ofthe peak D, and the dash line at point e may be the second boundary ofthe peak on the left side of the overlapping peak C, and the dash lineat the point g may be the second boundary of the peak on the right sideof the peak C.

Referring back to step 505, after the initial parameters are determined,the eddy-current field may be calibrated. In some embodiments, thevalues of the unknown parameters may cover a range of one or severalorders of magnitude. For instance, to various eddy-current components(e.g., those illustrated in FIG. 13), the values of k may cover one orseveral orders of magnitude. The determination of the unknown parametersmay be difficult using an unconstrained optimization.

In some embodiments, the value range of k_(l) may be determined by thetwo boundaries of a spectrum peak in the sparse Laplacian spectrum of xrelating to k, and the corresponding total intensity x0 _(l) may beclose to the true value of x_(l). In some embodiments, The boundary ofx_(l) may be set as xlb_(l)=μ·x0 _(l) and xub_(l)=v·x0 _(l). If x0 ₁>0,then 021 μ<1 and 1<v<+∞, and if x0 ₁<0, then 1<μ<+∞ and 0<v<1. Merely byway of example, the boundaries of x_(l) may be calculated by settingμ=0.5 and v=2(x0₁>0), or setting μ=2 and v=0.5 (x0₁<0). By applying theabove steps, the unknown parameters may fall within a range. In someembodiments, an appropriate selection of the initial parameters of theeddy-current field model may facilitate a global optimization. In someembodiments, an appropriate selection of the initial parameters mayfacilitate a local optimization. In some embodiments, an appropriateselection of the initial parameters may facilitate a local optimizationand a global optimization, and the optimization may be more robust. Asused herein, a global optimization may indicate that a set of parametersof the eddy-current field model provides a best solution of an objectivefunction (e.g., a minimized value of the objective function) in anentire parametric space. As used herein, a local optimization mayindicate that a set of parameters of the eddy-current field modelprovides a best solution of an objective function (e.g., a minimizedvalue of the objective function) in a subset of the entire parametricspace.

According to Equation (2), the estimation of the unknown parameters ofthe eddy-current field model may be calculated by the followingequation:

$\begin{matrix}{{\min\limits_{x,k}\{ {\sum_{m = l}^{M}( {d_{m} - {\sum_{l = 1}^{L}{x_{l}e^{{- T_{m}} \cdot k_{l}}}}} )^{2}} \}},} & (7)\end{matrix}$

where

${x_{l} = {\gamma \; \frac{G_{X}}{T_{A}}\alpha_{l}{\tau_{l}( {1 - e^{{- T_{A}}/\tau_{l}}} )}}},$

k_(l)=1/τ_(l), d_(m) is a eddy-current data of m, γ is the gyromagneticratio of proton, G_(x) is a testing gradient of the magnetic field,T_(A) is the duration of the falling edge of the testing gradient, andT_(m) is the d_(m) delay time of sampling the eddy-current data.m(m=1,2, . . . , M) represents the sequence number of sampling timing,l(l=1,2, . . . , L) represents the sequence number of the eddy-currentcomponent, α_(l) represents the amplitude of the eddy-current componentl, and τ_(l) represents the a characteristic time of an eddy-currentcomponent l. A testing gradient may be provide to determine aneddy-current that may be generated in response.

In some embodiments, the veracity of the compensation method may beimproved. For example, the improvement may be performed by utilizing abilateral linear inequality constrain to the eddy-current field modelbased on the peak information of the acquired reliable spectrum peaks.In some embodiments, the bilateral linear inequality constraint may beachieved using an interior point method. For example a logarithm barrierfunction may be added to the original objective function as a penaltyterm. The updated objective function may be defined as

$\begin{matrix}{{\min\limits_{\substack{{xlb}_{l} < x_{l} < {xub}_{l} \\ {klb}_{l} < k_{l} < {kub}_{l}}}\{ {{\sum_{m = l}^{M}( {d_{m} - {\sum_{l = 1}^{L}{x_{l}e^{{- T_{m}} \cdot k_{l}}}}} )^{2}} + {P( {x,k} )}} \}},} & (8)\end{matrix}$

where the penalty term P(x, k) may be defined as:

P(x, k)=−β·Σ_(l=1) ^(L)[log(x _(l) −xlb _(l))+log(xub _(l) −x_(l))+log(k _(l) −klb _(l))+log(kub_(l) −k _(l))].   (9)

In Equation (9), β may denote the Lagrange's multiplier, and its valuemay be determined by a minimum residual error. For example, the methodmay be achieved by a numerical method described in “Numerical Recipes,W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling,Cambridge University Press, New York, 1988,” the contents of which arehereby incorporated by reference.

The values of x and k may be calculated via the constrained optimizationdisclosed elsewhere in the present disclosure. The unknown parameter τof the eddy-current field model may be obtained by calculating thereciprocal of k. As mentioned above, the undetermined parameter a may becalculated according to x and τ. Both of the unknown parameters of theeddy-current field model may be calculated. A calibrated eddy-currentfield model may be obtained according to α and τ, and further used infuture eddy-current field compensation. When an MRI scanner is used inan operation, the operation parameters may be the same as during a testin which a calibrated eddy-current field model is determined. Theeddy-current field during the operation may be essentially the same asan eddy-current field during the test. As used herein, “essentially,” asin “essentially the same,” etc., with respect to a parameter or afeature may indicate that the variation is within 2%, or 5%, or 8%, or10%, or 15%, or 20% of the parameter or the feature. The eddy-currentfield during the operation may be compensated based on the correspondingcalibrated eddy-current field model. In some embodiments, at least onecharacteristic component of the eddy-current field may be identifiedbased on the calibrated eddy-current field model. An eddy-current fieldthat includes or may be described by the characteristic component may becompensated based on the at least one characteristic component of theeddy-current field. In some embodiments, a plurality of characteristiccomponents of the eddy-current field may be identified based on thecalibrated eddy-current field model. An eddy-current field that includesor may be described by the plurality of characteristic components may becompensated based on the plurality of characteristic components.

FIG. 12 depicts a curve of raw eddy-current and a curve of fittingeddy-current according to some embodiments of the present disclosure.The curve of raw eddy-current was obtained by measuring an actualeddy-current field. The raw data are illustrated as stars connected by acurve. The curve fitting of the raw eddy-current was obtained based on acalibrated compensation model obtained according to the system andmethod disclosed herein. As shown in FIG. 12, the curve based on thecalibrated compensation model may provide a close approximation of theraw (or actual) eddy-current field.

FIG. 13 depicts curve fitting of different components of an actualeddy-current field according to some embodiments of the presentdisclosure. In some embodiments, the curves of the different componentsmay be added together to obtain the curve of the eddy-current based onthe calibrated compensation model as illustrated in FIG. 12. The curvewith the circles was for the component of i=164 microseconds. The curvewith the squares was for the component of τ=37.4 microseconds. The curvewith the crosses was for the component of τ=6.12 microseconds. The curvewith the bars was for the component of τ=0.27 microseconds.

It should be noted that the above description of the three embodimentsare provided for a purpose of comprehending the present disclosure, notintended to limit the scope of the present disclosure. For personshaving ordinary skills in the art, various variations and modificationmay be conducted in the light of the present disclosure. However, thevariations and the modifications do not depart from the scope of thepresent disclosure.

Having thus described the basic concepts, it may be rather apparent tothose skilled in the art after reading this detailed disclosure that theforegoing detailed disclosure is intended to be presented by way ofexample only and is not limiting. Various alterations, improvements, andmodifications may occur and are intended to those skilled in the art,though not expressly stated herein. These alterations, improvements, andmodifications are intended to be suggested by this disclosure, and arewithin the spirit and scope of the exemplary embodiments of thisdisclosure.

Moreover, certain terminology has been used to describe embodiments ofthe present disclosure. For example, the terms “one embodiment,” “anembodiment,” and/or “some embodiments” mean that a particular feature,structure or characteristic described in connection with the embodimentis included in at least one embodiment of the present disclosure.Therefore, it is emphasized and should be appreciated that two or morereferences to “an embodiment” or “one embodiment” or “an alternativeembodiment” in various portions of this specification are notnecessarily all referring to the same embodiment. Furthermore, theparticular features, structures or characteristics may be combined assuitable in one or more embodiments of the present disclosure.

Further, it will be appreciated by one skilled in the art, aspects ofthe present disclosure may be illustrated and described herein in any ofa number of patentable classes or context including any new and usefulprocess, machine, manufacture, or composition of matter, or any new anduseful improvement thereof. Accordingly, aspects of the presentdisclosure may be implemented entirely hardware, entirely software(including firmware, resident software, micro-code, etc.) or combiningsoftware and hardware implementation that may all generally be referredto herein as a “block,” “module,” “engine,” “unit,” “component,” or“system.” Furthermore, aspects of the present disclosure may take theform of a computer program product embodied in one or more computerreadable media having computer readable program code embodied thereon.

A computer readable signal medium may include a propagated data signalwith computer readable program code embodied therein, for example, inbaseband or as part of a carrier wave. Such a propagated signal may takeany of a variety of forms, including electro-magnetic, optical, or thelike, or any suitable combination thereof. A computer readable signalmedium may be any computer readable medium that is not a computerreadable storage medium and that may communicate, propagate, ortransport a program for use by or in connection with an instructionexecution system, apparatus, or device. Program code embodied on acomputer readable signal medium may be transmitted using any appropriatemedium, including wireless, wireline, optical fiber cable, RF, or thelike, or any suitable combination of the foregoing.

Computer program code for carrying out operations for aspects of thepresent disclosure may be written in any combination of one or moreprogramming languages, including an object oriented programming languagesuch as Java, Scala, Smalltalk, Eiffel, JADE, Emerald, C++, C#, VB. NET,Python or the like, conventional procedural programming languages, suchas the “C” programming language, Visual Basic, Fortran 2003, Perl, COBOL2002, PHP, ABAP, dynamic programming languages such as Python, Ruby andGroovy, or other programming languages. The program code may executeentirely on the user's computer, partly on the user's computer, as astand-alone software package, partly on the user's computer and partlyon a remote computer or entirely on the remote computer or server. Inthe latter scenario, the remote computer may be connected to the user'scomputer through any type of network, including a local area network(LAN) or a wide area network (WAN), or the connection may be made to anexternal computer (for example, through the Internet using an InternetService Provider) or in a cloud computing environment or offered as aservice such as a Software as a Service (SaaS).

Furthermore, the recited order of processing elements or sequences, orthe use of numbers, letters, or other designations therefore, is notintended to limit the claimed processes and methods to any order exceptas may be specified in the claims. Although the above disclosurediscusses through various examples what is currently considered to be avariety of useful embodiments of the disclosure, it is to be understoodthat such detail is solely for that purpose, and that the appendedclaims are not limited to the disclosed embodiments, but, on thecontrary, are intended to cover modifications and equivalentarrangements that are within the spirit and scope of the disclosedembodiments. For example, although the implementation of variouscomponents described above may be embodied in a hardware device, it mayalso be implemented as a software only solution—e.g., an installation onan existing server or mobile device.

Similarly, it should be appreciated that in the foregoing description ofembodiments of the present disclosure, various features are sometimesgrouped together in a single embodiment, figure, or description thereoffor the purpose of streamlining the disclosure aiding in theunderstanding of one or more of the various inventive embodiments. Thismethod of disclosure, however, is not to be interpreted as reflecting anintention that the claimed subject matter requires more features thanare expressly recited in each claim. Rather, inventive embodiments liein less than all features of a single foregoing disclosed embodiment.

In some embodiments, the numbers expressing quantities of ingredients,properties such as molecular weight, reaction conditions, and so forth,used to describe and claim certain embodiments of the application are tobe understood as being modified in some instances by the term “about,”“approximate,” or “substantially.” For example, “about,” “approximate,”or “substantially” may indicate ±20% variation of the value itdescribes, unless otherwise stated. Accordingly, in some embodiments,the numerical parameters set forth in the written description andattached claims are approximations that may vary depending upon thedesired properties sought to be obtained by a particular embodiment. Insome embodiments, the numerical parameters should be construed in lightof the number of reported significant digits and by applying ordinaryrounding techniques. Notwithstanding that the numerical ranges andparameters setting forth the broad scope of some embodiments of theapplication are approximations, the numerical values set forth in thespecific examples are reported as precisely as practicable.

Each of the patents, patent applications, publications of patentapplications, and other material, such as articles, books,specifications, publications, documents, things, and/or the like,referenced herein is hereby incorporated herein by this reference in itsentirety for all purposes, excepting any prosecution file historyassociated with same, any of same that is inconsistent with or inconflict with the present document, or any of same that may have alimiting affect as to the broadest scope of the claims now or laterassociated with the present document. By way of example, should there beany inconsistency or conflict between the description, definition,and/or the use of a term associated with any of the incorporatedmaterial and that associated with the present document, the description,definition, and/or the use of the term in the present document shallprevail.

In closing, it is to be understood that the embodiments of theapplication disclosed herein are illustrative of the principles of theembodiments of the application. Other modifications that may be employedmay be within the scope of the application. Thus, by way of example, butnot of limitation, alternative configurations of the embodiments of theapplication may be utilized in accordance with the teachings herein.Accordingly, embodiments of the present application are not limited tothat precisely as shown and described.

1-23. (canceled)
 24. A system, comprising: at least one storage devicestoring executable instructions, and at least one processor incommunication with the at least one storage device, when executing theexecutable instructions, causing the system to perform operationsincluding: obtaining eddy-current data of an eddy-current field;performing a transformation operation on the eddy-current data of theeddy-current field to obtain transformed eddy-current data of theeddy-current field; determining, based on the transformed eddy-currentdata, one or more initial values of one or more eddy-current parametersof an eddy-current field model; determining, based on the one or moreinitial values of the one or more eddy-current parameters, one or moretarget values of the one or more eddy-current parameters; andcalibrating the eddy-current field based on the eddy-current field modelwith the one or more target values of the one or more eddy-currentparameters.
 25. The system of claim 24, wherein the transform operationincludes an inverse Laplace transformation and the transformed data ofthe eddy-current field includes a Laplacian spectrum.
 26. The system ofclaim 24, wherein to determine, based on the transformed eddy-currentdata, one or more initial values of one or more eddy-current parametersof an eddy-current field model, the at least one processor causes thesystem to perform additional operations including: determining, based onthe transformed eddy-current data, one or more initial values of one ormore first parameters associated with the transformed eddy-current data,each of the one or more first parameters corresponding to one of the oneor more eddy-current parameters associated with the eddy-current data;and determining, based on the initial values of the one or more firstparameters, the initial values of the one or more eddy-currentparameters.
 27. The system of claim 26, wherein to determine, based onthe transformed data of the eddy-current field, one or more initialvalues of one or more first parameters associated with the transformededdy-current data, the at least one processor causes the system toperform the operations including: acquiring information of a peak of theLaplacian spectrum; and determining the one or more initial values ofthe one or more first parameters based on the information of the peak.28. The system of claim 27, wherein the information of the peakcomprises at least one type of position, boundary, or intensity.
 29. Thesystem of claim 27, wherein to acquire information of a peak of theLaplacian spectrum, the at least one processor causes the system toperform the operations including: obtaining a first threshold; comparingthe absolute value of an intensity of the peak with the first threshold;and marking, if the absolute value of the intensity of the peak exceedsthe first threshold, the peak as a reliable spectrum peak.
 30. Thesystem of claim 27, wherein the peak comprises a reliable spectrum peak,and to determine the one or more initial values of the one or more firstparameters based on the information of the peak, the at least oneprocessor causes the system to perform the operations including:determining a boundary of the reliable spectrum peak; calculating atotal intensity of the reliable spectrum peak within the boundary; andsetting the one or more initial values of the one or more firstparameter based on the total intensity of the reliable spectrum peak.31. The system of claim 26, wherein to determine, based on the initialvalues of the one or more first parameters, the initial values of theone or more eddy-current parameters, the at least one processor causesthe system to perform the operations including: obtaining a relationshipbetween a value of each of the one or more first parameters and a valueof at least one of the one or more eddy-current parameters; anddetermining, based on the initial values of the one or more firstparameters and the relationship, the initial values of the one or moreeddy-current parameters, wherein the one or more eddy-current parametersinclude an amplitude and a characteristic time of each of one or moreeddy-current components included in the eddy-current field.
 32. Thesystem of claim 31, wherein the relationship is determined based on atransformation of the eddy-current field model.
 33. The system of claim24, wherein to determine, based on the one or more initial values of theone or more eddy-current parameters, one or more target values of theone or more eddy-current parameters, the at least one processor causesthe system to perform the operations including: determining, based onthe one or more initial values of the one or more eddy-currentparameters, the one or more target values of the one or moreeddy-current parameters using a least square algorithm.
 34. The systemof claim 33, wherein to determine, based on the one or more initialvalues of the one or more eddy-current parameters, the one or moretarget values of the one or more eddy-current parameters using the leastsquare algorithm, the at least one processor causes the system toperform the operations including: obtaining an objective functionassociated with the one or more eddy-current parameters, the objectivefunction denoting a difference between the eddy-current data and anestimated eddy-current data determined based on the one or more targetvalues of the one or more eddy-current parameters; and determining,based on the objective function and the one or more initial values ofthe one or more eddy-current parameters using the least squarealgorithm, the one or more target values of the one or more eddy-current35. A system, comprising: at least one storage device storing executableinstructions, and at least one processor in communication with the atleast one storage device, when executing the executable instructions,causing the system to perform operations including: obtainingeddy-current data of an eddy-current field; performing a transformationoperation on the eddy-current data of the eddy-current field to obtaintransformed eddy-current data of the eddy-current field; determining,based on the transformed eddy-current data of the eddy-current field,one or more target values of one or more first parameters associatedwith the transformed eddy-current data; determining, based on the one ormore target values of the one or more first parameters, one or moretarget values of one or more second parameters of the eddy-current data,wherein the one or more second parameters correspond to the one or morefirst parameters and include one or more eddy-current parameters;determining an eddy-current field model including the one or more targetvalues of the one or more second parameters; and calibrating theeddy-current data based on the eddy-current field model.
 36. The systemof claim 35, wherein to determine, based on the transformed eddy-currentdata of the eddy-current field, one or more target values of one or morefirst parameters associated with the transformed eddy-current data, theat least one processor causes the system to perform the operationsincluding: establishing an objective function associated with the one ormore first parameters associated with the transformed eddy-current dataof the eddy-current field, the objective function denoting a differencebetween the eddy-current data and an estimated eddy-current datadetermined based on the one or more target values of the one or morefirst parameters; and determining, based on the objective function andthe transformed eddy-current data, the one or more target values of theone or more first parameters.
 37. The system of claim 36, wherein todetermine, based on the objective function and the transformededdy-current data, the one or more first parameters, the at least oneprocessor causes the system to perform the operations including:determining, based on the transformed eddy-current data, one or moreinitial values of the one or more first parameters associated with thetransformed eddy-current data; and determining, based on the objectivefunction and the initial values of the one or more first parameters, theone or more target values of the one or more first parameters.
 38. Thesystem of claim 37, wherein the transform operation includes an inverseLaplace transformation and the transformed data of the eddy-currentfield includes a Laplacian spectrum.
 39. The system of claim38, whereinto determine, based on the transformed data of the eddy-current field,one or more initial values of the one or more first parametersassociated with the transformed eddy-current data, the at least oneprocessor causes the system to perform the operations including:acquiring information of a peak of the Laplacian spectrum; anddetermining the one or more initial values of the one or more firstparameters based on the information of the peak.
 40. The system of claim39, wherein the objective function includes a penalty term constructedbased on one or more logarithmic functions associated with the one ormore firsts parameters.
 41. The system of claim 35, wherein todetermine, based on the one or more target values of the one or morefirst parameters, one or more target values of the one or more secondparameters of the eddy-current data, the at least one processor causesthe system to perform the operations including: obtaining a relationshipbetween a value of each of the one or more first parameters and a valueof at least one of the one or more eddy-current parameters; anddetermining, based on the initial values of the one or more firstparameters and the relationship, the initial values of the eddy-currentparameters, wherein the eddy-current parameters includes an amplitudeand a characteristic time of each of one or more eddy-current componentsincluded in the eddy-current field.
 42. The system of claim 41, whereinthe relationship is determined based on the transform operation on theeddy-current field model.
 43. A method implemented on a computing devicehaving at least one processor and at least one storage device, themethod comprising: obtaining eddy-current data of an eddy-current field;performing a transformation operation on the eddy-current data of theeddy-current field to obtain transformed eddy-current data of theeddy-current field; determining, based on the transformed eddy-currentdata, one or more initial values of one or more eddy-current parametersof an eddy-current field model; determining, based on the one or moreinitial values of the one or more eddy-current parameters, one or moretarget values of the one or more eddy-current parameters; andcalibrating the eddy-current field based on the eddy-current field modelwith the one or more target values of the one or more eddy-currentparameters.